Chiral Liouville Gravity

TL;DR

This work constructs a chiral (warped) version of Liouville gravity in two dimensions by gauge-fixing to g_{--}=0 and ∂_-(g^{+-}g_{++})=0, which eliminates the left-moving Virasoro sector and yields a right-moving Virasoro–Kac-Moody symmetry with left translations encoded by Δ. Starting from the covariant Polyakov action, the authors derive a local chiral Liouville action for the fields ρ and h, establish the corresponding conserved currents, and develop a canonical (Dirac) formalism with explicit Dirac brackets that realize a Virasoro–Kac-Moody algebra with central charge c and level k=-4Δ. The paper demonstrates integrability via a Bäcklund transformation to free fields and discusses coupling to matter (e.g., a scalar field) that preserves warped conformal symmetry, outlining how matter modifies the currents and brackets. A program for quantization is discussed, emphasizing the need for regulators that preserve the chiral symmetry and suggesting avenues to relate the chiral theory to ordinary Liouville theory; the framework hints at holographically consistent chiral boundary dynamics in AdS_3 contexts. Overall, the work provides a semiclassical, integrable, warped-conformal gravity model with explicit charges and brackets, and lays groundwork for potential quantum realizations and holographic applications.

Abstract

Classical two-dimensional Liouville gravity is often considered in conformal gauge which has a residual left and right Virasoro symmetry algebra. We consider an alternate, chiral, gauge which has a residual right Virasoro Kac-Moody algebra, and no left Virasoro algebra. The Kac-Moody zero mode is the left-moving energy. Dirac brackets of the constrained Hamiltonian theory are derived, and the residual symmetries are shown to be generated by integrals of the conserved chiral currents. The central charge and Kac-Moody level are computed. The possible existence of a corresponding quantum theory is discussed.